Problem: Compute $\binom{17}{9}$. You are told that $\binom{15}{6} = 5005$ and $\binom{15}{8} = 6435$.
By Pascal's Rule, \begin{align*}
\binom{17}{9} &= \binom{16}{9} + \binom{16}{8} \\
\binom{17}{9} &= \binom{15}{8} + \binom{15}{9} + \binom{15}{7} + \binom{15}{8} \\
\binom{17}{9} &= \binom{15}{8} + \binom{15}{15-9} + \binom{15}{15-7} + \binom{15}{8} \\
\binom{17}{9} &= \binom{15}{8} + \binom{15}{6} + \binom{15}{8} + \binom{15}{8} \\
\binom{17}{9} &= 6435 + 5005 + 6435 + 6435 \\
\binom{17}{9} &= \boxed{24310}
\end{align*}